First, make the vinaigrette: place all of the vinaigrette ingredients in a medium bowl and whisk until emulsified. Alternatively, place all the ingredients in a mason jar, close tightly, and shake vigorously until emulsified. Set aside until needed. The vinaigrette can be made well in advance and will keep, refrigerated, for up to a week.
Next, prepare the Brussels sprouts. Trim the stem end of each sprout, peel off any wilted outer leaves, and thinly slice either with a knife or the slicing attachment on your food processor.
Place the shredded Brussels sprouts in a large bowl, drizzle with the vinaigrette, and toss together until everything is evenly coated. Sprinkle with the cherries and pistachios, briefly toss together again, and serve immediately.